The use of adomian decomposition method for solving the regularized long-wave equation

Chaos, Solitons and Fractals 26 (2005) 747–757 www.elsevier.com/locate/chaos

The use of adomian decomposition method for solving the regularized long-wave equation Talaat S. El-Danaf, Mohamed A. Ramadan *, Faysal E.I. Abd Alaal Department of Mathematics, Faculty of Science, Menoufia University, Shiben El-Kom, Egypt Accepted 28 January 2005

Communicated by Prof. M. Wadai

Abstract In this paper, an accurate method to obtain an approximate numerical solution for the nonlinear regularized longwave (in short RLW) equation is considered. The theoretical analysis of the method is investigated. The performance and the accuracy of the algorithm are illustrated by solving two test examples of the problem. The obtained results are presented and compared with the analytical solutions. It is observed that only few terms of the series expansion are required to obtain approximate solutions with good accuracy.
2005 Elsevier Ltd. All rights reserved.

1. Introduction The regularized long wave (RLW) equation is defined as ut þ ux þ auux
buxxt ¼ 0:

ð1Þ

It is a different explanation of nonlinear dispersive waves to the more familiar Korteweg–de Vries (KDV) equation of the form ut þ ux þ uux þ uxxx ¼ 0:

ð2Þ

The RLW equation, at first, proposed by Peregrine [3]. This equation plays a major role in the study of nonlinear dispersive waves [3,7] because of its description to a larger number of important physical phenomena, such as shallow water waves and ion acoustic plasma waves. In this paper we apply Adomian decomposition method (ADM) for the RLW equation. The ADM is a numerical technique for solving linear or nonlinear, algebraic or ordinary/partial differential equations. It is closely related to the

*

Corresponding author. E-mail address: [email protected] (M.A. Ramadan).

0960-0779/$ - see front matter
2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.02.012

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Taylor series and fixed-point iteration methods. The ADM, see for example [2,4] provides the numerical solutions of differential and integral equations by generating a functional series solution in a very efficient manner. The obtained series may provide the solution in a closed form. However, for concrete problems, the n-term approximation /n defined by: /n ¼

n
1 X

uk ðx; tÞ

ð3Þ

k¼0

can be used for numerical approximation. It was shown in [2,4,5] among others that the effectiveness of the method can be dramatically improved by determining further components of the solution u(x, t). The reliability of the ADM gives a wider applicability in handling evaluation models.

2. Analysis of the method For the purpose of illustration of the methodology to the proposed method, using the ADM, we begin by considering Eq. (1) in the operator form: Lu þ RðuÞ þ N ðuÞ ¼ 0;

ð4Þ

where LðL ¼ oto Þ is a linear operator and R its remainder of the linear operator. The nonlinear term is represented by N(u). Thus we get Lu ¼ 0
RðuÞ
N ðuÞ:

ð5Þ

Assuming the inverse operator L
1 exists and it can be taken as the definite integral with respect to t from t0 to t, i.e.! L
1 ¼

Z

t

ð:Þdt

ð6Þ

t0

then, applying the inverse operator L
1 on both sides of Eq. (5) yields u ¼ f0 þ L
1 ð0
RðuÞ
N ðuÞÞ;

ð7Þ

where f0 is the solution of homogeneous equation Lu ¼ 0

ð8Þ

involving the constants of integration. The integration constants that involved in the solution of homogeneous equation (8) are to be determined by the initial condition u(x, 0) = u0 = f(x). The ADM assumes that the unknown function u(x, t) can be expressed by a sum of components defined by the decomposition series of the form: 1 X uðx; tÞ ¼ un ðx; tÞ ð9Þ n¼0

with u0 defined as u(x, 0) where u(x, t) will be determined recursively. The nonlinear operator N(u) in Eq. (5) can also be decomposed by an infinite series of polynomials given by 1 X An ðu0 ; u1 ; . . . ; un Þ; ð10Þ N ðuÞ ¼ n¼0

where An(u0 ,u1, u2, . . ., un) are the appropriate AdomianÕs polynomials. These polynomials are defined as: " !# 1 X 1 dn i An ¼ N k ui n! dkn i¼0

: k¼0

ð11Þ

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It is known in the literature that these polynomials can be calculated for all forms of nonlinearity according to algorithms constructed by Adomian [4,5] and recently developed by an alternative approach, see for example, [1,6]. From the above theoretical analysis one can arrive to the following remark. Remark. The solution of the nonlinear PDE equation (1) which is rewritten in the operator form equtaion (4) with the initial condition u(x, 0) = u0 = f(x), can be determined by the series equation (9) with the iterative process: uðx; 0Þ ¼ f ðxÞ; unþ1 ðx; tÞ ¼ L
1 ð0
Rðun Þ
An Þ;

ð12Þ

n is integer:

3. Application and numerical results Applying the inverse operator L
1 on both sides of equation (1) we get: uðx; tÞ ¼ f ðxÞ
L
1 ðauux þ ux
buxxt Þ:

ð13Þ

Now, by using Eqs. (9)–(11) and (13) we obtain: 1 X


1

un ðx; tÞ ¼ f ðxÞ
L

a

n¼0

1 X

An þ

n¼0

1 X n¼0

!
b

un x

1 X n¼0

! ! un

:

ð14Þ

xxt

Identifying the zeroth component u0(x, t) as f(x), the remaining components un(x, t),n P 1 can be determined by using the recurrence relation Eq. (12). That is, u0 ðx; tÞ ¼ f ðxÞ; unþ1 ðx; tÞ ¼
L
1 ðaAn þ ðun Þx
bðun Þxxt Þ;

n P 0;

ð15Þ

where An are AdomianÕs polynomials that represent the nonlinear term uux. One can see that the first few terms of An are given by: A0 ¼ u0x u0 ; A1 ¼ u0x u1 þ u1x u0 ; A2 ¼ u0x u2 þ u1x u1 þ u2x u0 ;

ð16Þ

A3 ¼ u0x u3 þ u1x u2 þ u2x u1 þ u3x u0 : The remaining polynomials can be generated in a similar way. The scheme in Eq. (15) can easily determine the components un(x, t),n P 0 and the first few components of un(x, t) follows immediately upon setting u0 ðx; tÞ ¼ f ðxÞ; u1 ðx; tÞ ¼
L
1 ðaA0 þ ðu0 Þx
bðu0 Þxxt Þ; u2 ðx; tÞ ¼
L
1 ðaA1 þ ðu1 Þx
bðu1 Þxxt Þ;

ð17Þ


1

u3 ðx; tÞ ¼
L ðaA2 þ ðu2 Þx
bðu2 Þxxt Þ; u4 ðx; tÞ ¼
L
1 ðaA3 þ ðu3 Þx
bðu3 Þxxt Þ: It is in principle, possible to calculate more components in the decomposition series P to enhance the approximation. Consequently, one can recursively determine each individual term of the series 1 n¼0 un ðx; tÞ, and hence the solution u(x, t) is readily obtained in a series form. It is interesting to note that we obtain the solution by using the initial condition u0(x, t) = f(x) only.

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The accuracy of the proposed method, based on the ADM, is investigated by considering two test examples for the RLW equation. The obtained numerical approximate solution uappr.(x, t) for each example is compared with the exact solution where uappr: ðx; tÞ ¼ u0 ðx; tÞ þ u1 ðx; tÞ þ u2 ðx; tÞ þ u3 ðx; tÞ þ u4 ðx; tÞ:

ð18Þ

Example 1. Consider the RLW equation (1) for a = b = 1 with the boundary conditions uð
25; tÞ ¼ uð25; tÞ ¼ 0;

ð19Þ

tP0

and the initial condition uðx; 0Þ ¼ 3c sec h2 ðpxÞ;

ð20Þ

where c is an arbitrary constant and p = (1/2)(c/(1 + c))1/2. The single solitary-wave solution of RLW equation [3] is given by: uðx; tÞ ¼ 3c sec h2 ðpx
xt þ /Þ;

ð21Þ

where x = (1/2)(c(1 + c))1/2 and / is an arbitrary constant. The obtained numerical results are summarized in Tables 1–7. From these results we conclude that the method, i.e. to calculate the approximate numerical solution of this RLW equation, gives remarkable accuracy in comparison with the analytical solution especially for small values of time t. Two different values for the arbitrary constant c are considered where we take different values of the time t for each constant c.

Table 1 For constant c = 0.03 and time t = 0.4 x

Approx. solution

Exact solution

Absolute error


25
15
5 5 15 25

0.004586905240735977 0.02257104019906218 0.07323808486739658 0.07750201785258855 0.025459695860876534 0.0052589932303313885

0.004586904594327663 0.022571052966962456 0.07323773676637371 0.07750150990598588 0.025459764131365713 0.005258986212585601

6.46408 · 10
10 1.27679 · 10
8 3.48101 · 10
7 5.07947 · 10
7 6.82705 · 10
8 7.01775 · 10
9

Table 2 For constant c = 0.03 and time t = 1 x

Approx. solution

Exact solution

Absolute error


25
15
5 5 15 25

0.0041384454841287876 0.02058777501040447 0.06982310494902395 0.08041695840825432 0.027815620621058787 0.005824795055136458

0.004138448123927186 0.02058782092050679 0.06982313070350868 0.08041301459062627 0.02781578877036916 0.005824729708230179

2.6398 · 10
9 4.59101 · 10
8 2.57545 · 10
8 3.94382 · 10
6 1.68149 · 10
7 6.53469 · 10
8

Table 3 For constant c = 0.03 and time t = 2.5 x

Approx. solution

Exact solution

Absolute error


25
15
5 5 15 25

0.003196598430493901 0.01627426710845436 0.06080176654997385 0.0862554782644622 0.0344408570119406 0.007508247311437174

0.003196459547901543 0.0162721309757194 0.06083746191832978 0.08625359480333179 0.03443235952215925 0.007507381186451155

1.38883 · 10
7 2.13613 · 10
6 3.56954 · 10
5 1.88346 · 10
6 8.49749 · 10
6 8.66125 · 10
7

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Next, we draw some of the obtained approximate solutions uappr. (x, t) for this first test problem (for different values of the arbitrary constant c and the time t) versus the distance x. The following figures show the behavior of the numerical solution for the first test problem.

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Example 2. The RLW equation has the analytic solution form [7,8]: uðx; tÞ ¼ d þ 3c sec h2 ðk½ðx
rÞ
ðd
cÞtÞ; qffiffiffiffiffiffiffiffiffiffi ac with k ¼ 12 bðdþcÞ , where d and c are constants.

x 2 ½0; 80

ð22Þ

T.S. El-Danaf et al. / Chaos, Solitons and Fractals 26 (2005) 747–757

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Table 4 For constant c = 0.03 and time t = 3.5 x

Approx. Solution

Exact solution

Absolute error


25
15
5 5 15 25

0.0026900972061389946 0.013865995297049086 0.05465623505044572 0.08864195635698928 0.03945438256489056 0.008879645564938128

0.0026889490680928716 0.013860282346209738 0.05476742707917842 0.08873664380421509 0.03941025356509516 0.008877300085800517

1.14814 · 10
6 5.71295 · 10
6 1.11192 · 10
4 9.46874 · 10
5 4.4129 · 10
5 2.34548 · 10
6

Table 5 For constant c = 0.1 and time t = 0.4 x

Approx. solution

Exact solution

Absolute error


25
15
5 5 15 25

0.0005591668802803966 0.01119892439741671 0.16321335208867324 0.19338378604435588 0.01452007407451283 0.0007288450637729317

0.000559168528082687 0.011198838972354058 0.16325976076217727 0.19331405594840334 0.01451848762035994 0.0007288716093509047

1.6478 · 10
9 8.54251 · 10
8 4.64087 · 10
5 6.97301 · 10
5 1.58645 · 10
6 2.65456 · 10
8

For computational work, we choose a = b = 1 and d = 1. The obtained numerical results are summarized in Tables 8–13 where the approximate numerical solution of this RLW equation is compared with the analytical solution. One can see that for the two arbitrary considered values of c used with different values of the time t the errors are acceptable. The following figures show the behavior of numerical solution for the second test problem where the y-axis represents the numerical solution uappr.(x, t) and x-axis represents the distance x.

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Table 6 For constant c = 0.1 time and t = 1 x

Approx. solution

Exact solution

Absolute error


25
15
5 5 15 25

0.0004583510950516169 0.00920988494264213 0.14218403295207901 0.21602532485793088 0.01763017419446828 0.0008887699312082724

0.00045834477066029194 0.009209315833411044 0.14200233387991465 0.21623246785781725 0.01761998467946689 0.0008891157860494625

6.32439 · 10
9 5.69109 · 10
7 1.81699 · 10
4 2.07143 · 10
4 1.01895 · 10
5 3.45855 · 10
7

Table 7 For constant c = 0.1 and time t = 2.5 x

Approx. solution

Exact solution

Absolute error


25
15
5 5 15 25

0.00027869238130987856 0.005629244308911319 0.09867557823625717 0.2585164176359685 0.02851457773669314 0.0014519797439721083

0.0002787812726255066 0.005633854745967372 0.09643346122513724 0.26796773636514587 0.028428813854319734 0.0014608379814321855

8.88913 · 10
8 4.61044 · 10
6 2.24212 · 10
3 9.45132 · 10
3 8.57639 · 10
5 8.85824 · 10
6

Table 8 For constant c = 0.03, r = 40 and time t = 0.5 x

Approx. solution

Exact solution

Absolute error

0 20 40 60 80

1.000326923470031 1.0094250096522948 1.0893589126147474 1.013071988354159 1.0004642599093592

1.0003587156146299 1.0102812160590833 1.0898460235452112 1.0120046689915063 1.0004231450040417

3.17921 · 10
5 8.56206 · 10
4 4.87111 · 10
4 1.06732 · 10
3 4.11149 · 10
5

Table 9 For constant c = 0.03, r = 40 and time t = 1.5 x

Approx. solution

Exact solution

Absolute error

0 20 40 60 80

1.0002301734664407 1.0067458413625132 1.0845386478570564 1.0179527218848878 1.0006587503392455

1.0003040798014746 1.008791392931373 1.0886267634545375 1.013991384205138 1.0004991144132063

7.39063 · 10
5 2.04555 · 10
3 4.08812 · 10
3 3.96134 · 10
3 1.59636 · 10
4

Table 10 For constant c = 0.03, r = 40 and time t = 2 x

Approx. solution

Exact solution

Absolute error

0 20 40 60 80

1.0001932491404137 1.0056979993593727 1.0807707162503926 1.0209507752087847 1.0007840137847777

1.0002799615943592 1.0081253161746522 1.0875779347472503 1.0150930620053165 1.0005420551845194

8.67125 · 10
5 2.42732 · 10
3 6.80722 · 10
3 5.85771 · 10
3 2.41959 · 10
4

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Table 11 For constant c = 0.1, r = 40 and time t = 0.5 x

Approx. solution

Exact solution

Absolute error

0 20 40 60 80

1.0000049814348688 1.0020651215480199 1.2924693947220136 1.0039897707234937 1.0000096655217574

1.0000060598686609 1.0025092266955649 1.2986235433433275 1.0032871757893336 1.0000079489924776

1.07843 · 10
6 4.44105 · 10
4 6.15415 · 10
3 7.02595 · 10
4 1.71653 · 10
6

Table 12 For constant c = 0.1, r = 40 and time t = 1 x

Approx. solution

Exact solution

Absolute error

0 20 40 60 80

1.0000035750487313 1.0014835529620931 1.272894190458302 1.0055264803625965 1.0000134274270351

1.0000052910107895 1.0021920261657724 1.2945443513157584 1.0037618494663754 1.000009104086404

1.71596 · 10
6 7.08473 · 10
4 2.16502 · 10
2 1.76463 · 10
3 4.32334 · 10
6

Table 13 Fore constant c = 0.1, r = 40 and time t = 1.5 x

Approx. solution

Exact solution

Absolute error

0 20 40 60 80

1.00000256522364 1.001065243647678 1.2503242219196093 1.0076137073723228 1.0000185543024147

1.0000046197025363 1.0019147951357663 1.2879091109295262 1.0043045692035322 1.0000104270278654

2.05448 · 10
6 8.49551 · 10
4 3.75849 · 10
2 3.30914 · 10
3 8.12727 · 10
6

4. Concluding remarks In this paper, we considered a numerical treatment for the solution of the RLW equation using ADM. From the theoretical analysis and the numerical results, we may come to the following concluding remarks. 1. We can claim that Adomian decomposition method is used successfully for solving the considered PDE. 2. From the outlined theoretical analysis, we can conclude that the proposed method is applicable for similar physical equations. 3. The obtained numerical results compared with the analytical solution show that the method provides remarkable accuracy especially for small values of the time t. The accuracy can be further improved by considering more terms in the series expansion. 4. The analytical and numerical solutions are obtained using initial conditions u(x, 0) = f(x) only. A Mathematica program is written and run on a PC Pentium IV.

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